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arXiv:2604.05216 · Physical Review A

Approximate vortex lattices of atomic Fermi superfluid on a spherical surface

Keshab Sony, Yan He, Chih-Chun Chien

cond-mat.quant-gascond-mat.supr-conquant-phSubmitted Apr 6, 2026• Announced Apr 8, 2026• 0 citations
v1 · Apr 6, 2026v2 · Oct 14, 2026
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Abstract

While planar Fermi superfluids form Abrikosov vortex lattices under magnetic or effective gauge fields, spherical geometry forbids perfect lattices above 20 vortices. We characterize approximate vortex structures of atomic Fermi superfluids under an effective monopole field on a spherical surface as an analogue of the planar vortex-lattice problem by two constructions based on the Ginzburg-Landau theory. The first one is geometric and uses the random, geodesic-dome, and Fibonacci lattices as scaffolds to construct the order parameter from the degenerate monopole harmonics. The second one minimizes the free energy by numerically adjusting the coefficients to find the solution with the minimal Abrikosov parameter. We have verified the vortices from both constructions are zeros of the order parameter with circulating currents around the vortex cores. As the number of vortices increases, the Abrikosov parameters of both the Fibonacci-lattice and minimization solutions extrapolate to the planar value. We briefly discuss implications for ultracold atoms in thin spherical-shell geometry.

Comments
9 pages, 5 figures, submitted

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While planar Fermi superfluids form Abrikosov vortex lattices under magnetic or effective gauge fields, spherical geometry forbids perfect lattices above 20 vortices. We characterize approximate vortex structures of atomic Fermi superfluids under an effective monopole field on a spherical surface as an analogue of the planar vortex-lattice problem by two constructions based on the Ginzburg-Landau theory. The first one is geometric and uses the random, geodesic-dome, and Fibonacci lattices as scaffolds to construct the order parameter from the degenerate monopole harmonics. The second one minimizes the free energy by numerically adjusting the coefficients to find the solution with the minimal Abrikosov parameter. We have verified the vortices from both constructions are zeros of the order parameter with circulating currents around the vortex cores. As the number of vortices increases, the Abrikosov parameters of both the Fibonacci-lattice and minimization solutions extrapolate to the planar value. We briefly discuss implications for ultracold atoms in thin spherical-shell geometry.
The structures of multiply quantized vortices (MQVs) of an equal-population atomic Fermi superfluid in a rotating spherical bubble trap approximated as a thin shell are analyzed by solving the Bogoliubov--de Gennes (BdG) equation throughout the Bardeen-Cooper-Schrieffer (BCS)--Bose-Einstein condensation (BEC) crossover. Consistent with the Poincar\'e-Hopf theorem, a pair of vortices emerge at the poles of the rotation axis in the presence of azimuthal symmetry and the compact geometry provides confinement for the MQVs. While the single-vorticity vortex structure is similar to that in a planar geometry, higher-vorticity vortices exhibit interesting phenomena at the vortex center, such as a density peak due to accumulation of a normal Fermi gas and reversed circulation of current due to in-gap states carrying angular momentum in the BCS regime but not the BEC regime because of the subtle relations between the order parameter and density. The energy spectrum shows the number of the in-gap state branches corresponds to the vorticity of a vortex, and an explanation based on a topological correspondence is provided.
We consider a superfluid of trapped fermionic atoms and study the single vortex solution in the Ginzburg-Landau regime. We define simple analytical estimates for the main characteristics of the system, such as the vortex core size, temperature regimes for the existence of a vortex, and the effects of rotation and interactions with normal fermions. The parameter dependence of the vortex core size (healing length) is found to be essentially different from that of the healing length in metallic superconductors or in trapped atomic Bose-Einstein condensation in the Thomas-Fermi limit. This is an indication of the importance of the confining geometry for the properties of fermionic superfluids.
In order to help detecting superfluidity, we theoretically investigate p-wave pairing superfluids in neutral Fermion atom gases confined by a three dimensimentional (3D) harmonic potential. The Ginzburg-Landau framework, which is generic for p-wave superfluids, is used to describe the order parameter spatial structure, or texture characterized by the l-vector both at rest and under rotation. The l-vector configuration is strongly contrained by the boundary condition due to a trap. It is found that the ground state textures exhibit spontaneous supercurrent at rest both cigar and pancake shape traps. The current direction depends on the trapping shape. Under rotation a pair of half-quantum vortex with half-winding number enters a system and is stabilized for both trap geometries. We give detailed explanation for their 3D structure. The deformations of the condensate shape are seen with increasing the rotation speed, which is tightly connected with the underlying vortex formation where the condensates are depressed in the vortex core.
The experimental realization of quantum-degenerate Bose gases made of atoms with sizeable magnetic dipole moments has created a new type of fluid, known as a quantum ferrofluid, which combines the extraordinary properties of superfluidity and ferrofluidity. A hallmark of superfluids is that they are constrained to rotate through vortices with quantized circulation. In quantum ferrofluids the long-range dipolar interactions add new ingredients by inducing magnetostriction and instabilities, and also affect the structural properties of vortices and vortex lattices. Here we give a review of the theory of vortices in dipolar Bose-Einstein condensates, exploring the interplay of magnetism with vorticity and contrasting this with the established behaviour in non-dipolar condensates. We cover single vortex solutions, including structure, energy and stability, vortex pairs, including interactions and dynamics, and also vortex lattices. Our discussion is founded on the mean-field theory provided by the dipolar Gross-Pitaevskii equation, ranging from analytic treatments based on the Thomas-Fermi (hydrodynamic) and variational approaches to full numerical simulations. Routes for generating vortices in dipolar condensates are discussed, with particular attention paid to rotating condensates, where surface instabilities drive the nucleation of vortices, and lead to the emergence of rich and varied vortex lattice structures. We also present an outlook, including potential extensions to degenerate Fermi gases, quantum Hall physics, toroidal systems and the Berezinskii-Kosterlitz-Thouless transition.
By implementing the Bogoliubov-de Gennes (BdG) formalism of population-imbalanced atomic Fermi gases with pairing interactions in a thin spherical shell, we characterize the Larkin-Ovchinnikov-Fulde-Ferrell (LOFF) state in such a compact geometry. We first construct a phase diagram showing where uniform solutions of spin-polarized Fermi superfluid from the BdG equation cease to exist due to the vanishing order parameter. Near the boundary, various LOFF states with spatially modulating order parameters and density profiles can survive as convergent solutions to the BdG equation. When both uniform and LOFF solutions are present, we compare their grand potentials to determine the energetically favorable state and find that the LOFF states with multiple nodes in the order parameter become more stable at higher spin polarization. However, the LOFF state only survives close to the phase boundary where the uniform solutions vanish, indicating fragility of the LOFF state on a spherical surface. We also briefly discuss possible implications.
In this paper, we study the structure of vortex lattices in a trapped superfluid Fermi gas in the unitary region near the superfluid transition temperature. A phenomenological approach based on the Ginzburg-Landau theory is used. Two types of stable triangular vortex lattices are found numerically at the same rotation frequency. In type (A), the trap center is occupied by a vortex; in type (B), there are three vortices closest to the trap center which is unoccupied. The lattice structures at different rotation frequencies and temperatures are analyzed. The vortex density increases linearly with the rotation frequency, very close to the analytical estimation. The superfluid and type-(B) vortex lattice do not disappear together. At high rotation frequencies, the vortex lattice disappears, but there exists a localized superfluid state near the trap center. The superfluid vanishes at a critical rotation frequency consistent with the theory for the upper critical magnetic field of type-II superconductors.